If you just passed the Tech, General, or Extra exam and you don’t have a call yet, you will need to wait until your new call is in the FCC database before you apply to change it.
Then, if you want to start searching for a call you like, here is the link to the FCC database with ALL the 700,000 licensed hams in the US. DATABASE
Now, to apply for a Vanity call:
1) Go to the FCC ULS
2) Log in with your FRN and Password - Submit
3) On the far right side - Click on Request Vanity Call Sign
4) Exempt from Fees – no, no - Continue
5) Primary Station preference list – Continue
6) Fill in the preference list – 1 to 25. You don’t have to ask for more than 1, but it doesn’t hurt to ask for several, up to 25. They will give you the first one on the list that works. – Continue
7) Ignore the warning about Administrative Name Change – Address change: fix it, otherwise Continue
8) Answer the Felony question – Continue
9) Check the Summary. Note: No fees are required. THIS WILL CHANGE SOON! - "You have submitted information in all the categories below. Use the view and edit capabilities to review your application to confirm that it is complete and accurate. Correct information as necessary. Once you are confident that the application is ready for certification and submission, click on the "Continue to Certify" button."
That’s it. The usual wait time if there is no conflict is about two weeks. Maybe less.